The proposer intends to undertake a mathematical study of several systems of differential equations which model the conduction of a nerve impulse. These models include the Hodgkin-Huxley equations, Nagumo's equation, Fitzhugh's equation, Hoyt's equation, and Noble's modification of the Hodgkin-Huxley equations. The proposer has found that under current clamp conditions there occurs a bifurcation of periodic solutions in the Hodgkin- Huxley and the Fitzhugh equations. Using the Hopf theory of bifurcation, the proposer intends to investigate Hoyt's equation and Noble's equations for the bifurcation, direction and stability of families of small periodic solutions. The shooting method has been successfully applied to Nagumo's equation by Hastings to prove the existence of a large periodic solution. The proposer intends to apply this method and fixed point theorems to the other mentioned models to obtain the existence of large periodic solutions. Finally, the global bifurcation theory of Yorke and Alexander has been successfully applied by the proposer to the Fitzhugh equations. It was proved that a family of small periodic solutions bifurcating from the equilibrium solution grows to become a large periodic solution corresponding to an infinite sequence of action potentials. This approach will be applied to the Hodgkin-Huxley, Hoyt, and Noble models to determine whether families of small periodic solutions bifurcating from the equilibrium solution will grow to become a large periodic solution.